Intersection Bodies in R4

AbstractIntersection bodies were introduced in 1988 by Lutwak, who found a close connection between those bodies and the well-known 1956 Busemann–Petty problem, which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume. The author has recently shown that an origin symmetric star bodyKis an intersection body if and only if the function ‖x‖−1Kis a positive definite distribution, where ‖x‖K=min{a⩾0:x∈aK}. We use this result to prove that the unit balls of the spaces ℓ4q, 2<q⩽∞, are intersection bodies. Forn⩾5 the unit balls of the spaces ℓnq, 2